18. Crovisier, J. et al. The spectrum of Comet Hale-Bopp (C/1995 01) observed with the Infrared Space

Observatory at 2.9 AU from the Sun. Science 275, 1904–1907 (1997).

19. Molster, F. J., Waters, L. B. F. M., Tielens, A. G. G. M., Koike, C. & Chihara, H. Crystalline silicate dust

around evolved stars. III. A correlations study of crystalline silicate features. Astron. Astrophys. 382,

241–255 (2002).

20. Malfait, K. et al. The spectrum of the young star HD 100546 observed with the Infrared Space

Observatory. Astron. Astrophys. 332, L25–L28 (1998).

21. Grady, C. A. et al. Infalling planetesimals in pre-main stellar systems. In Protostars and Planets IV (eds

Mannings, V., Boss, A. P. & Russell, S. S.) 613–638 (Univ. Arizona Press, 2000).

22. Acke, B. & van den Ancker, M. E. ISO spectroscopy of disks around Herbig Ae/Be stars. Astron.

Astrophys.(in the press); preprint at khttp://xxx.lanl.gov/astro-ph/0406050l (2004).23. Wetherill, G. W. Formation of the earth. Annu. Rev. Earth Planet. Sci. 18, 205–256 (1990).

24. Dorschner, J., Begemann, B., Henning, T., Jager, C. & Mutschke, H. Steps toward interstellar silicate

mineralogy. Astron. Astrophys. 300, 503–520 (1995).

25. Servoin, J. L. & Piriou, B. Infrared reflectivity and Raman scattering of Mg2SiO4 single crystal. Phys.

Status Solidi 55, 677–686 (1973).

26. Jager, C. et al. Steps toward interstellar silicate mineralogy. IV. The crystalline revolution. Astron.

Astrophys. 339, 904–916 (1998).

27. Spitzer, W. G. & Kleinman, D. A. Infrared lattice bands of quartz. Phys. Rev. 121, 1324–1335 (1961).

28. Min, M., Hovenier, J. W. & de Koter, A. Shape effects in scattering and absorption by randomly

oriented particles small compared to the wavelength. Astron. Astrophys. 404, 35–46 (2003).

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far-infrared region. Publ. Astron. Soc. Jpn 53, 243–250 (2001).

Supplementary Information accompanies the paper on www.nature.com/nature.

Acknowledgements The data is based on observations obtained at the European Southern

Observatory (ESO), Chile. We thank all those involved in building VLTI and MIDI. We thank

V. Icke for providing the illustration shown in Fig. 2. C.P. Dullemond is acknowledged for many

discussions.

Competing interests statement The authors declare that they have no competing financial

interests.

Correspondence and requests for materials should be addressed to R.v.B.

(vboekel@science.uva.nl).

..............................................................

Experimental demonstration ofquantum memory for lightBrian Julsgaard1, Jacob Sherson1,2, J. Ignacio Cirac3,Jaromır Fiurasek4 & Eugene S. Polzik1

1Niels Bohr Institute, Danish Quantum Optics Center – QUANTOP, CopenhagenUniversity, Blegdamsvej 17, 2100 Copenhagen Ø, Denmark2Department of Physics, Danish Quantum Optics Center – QUANTOP,University of Aarhus, 8000 Aarhus C, Denmark3Max Planck Institute for Quantum Optics, Hans-Kopfermann-Str. 1, Garching,D-85748, Germany4QUIC, Ecole Polytechnique, CP 165, Universite Libre de Bruxelles, 1050 Brussels,Belgium, and Department of Optics, Palacky University, 17. listopadu 50,77200 Olomouc, Czech Republic.............................................................................................................................................................................

The information carrier of today’s communications, a weak pulseof light, is an intrinsically quantum object. As a consequence,complete information about the pulse cannot be perfectlyrecorded in a classical memory, even in principle. In the fieldof quantum information, this has led to the long-standingchallenge of how to achieve a high-fidelity transfer of an inde-pendently prepared quantum state of light onto an atomicquantum state1–4. Here we propose and experimentally demon-strate a protocol for such a quantum memory based on atomicensembles. Recording of an externally provided quantum state oflight onto the atomic quantum memory is achieved with 70 percent fidelity, significantly higher than the limit for classicalrecording. Quantum storage of light is achieved in three steps:first, interaction of the input pulse and an entangling field withspin-polarized caesium atoms; second, subsequent measurementof the transmitted light; and third, feedback onto the atoms usinga radio-frequency magnetic pulse conditioned on the measure-

ment result. The density of recorded states is 33 per cent higherthan the best classical recording of light onto atoms, with aquantum memory lifetime of up to 4 milliseconds.

Light is a natural carrier of information in both classical andquantum communications. In classical communications, bits areencoded in large average amplitudes of light pulses, which aredetected, converted into electric signals and subsequently storedas charges or magnetization of memory cells. In quantum infor-mation processing, information is encoded in quantum states thatcannot be accurately recorded by such classical means. Consider astate of light defined by its amplitude and phase, or equivalentlyby two quadrature phase operators, XL and PL, with the canonicalcommutation relation [XL,PL] ¼ i. These variables play the samerole in quantum mechanics as the classical quadratures X, P do inthe decomposition of the electric field of light with the frequency qas E / X cosqt þ P sinqt. Other quantum properties of light, suchas the photon number n ¼ 1 X2

L þ P2L 2 1

� �; and so on, can be

expressed in terms of XL and PL.The best classical approach to recording a state of light onto

atoms would involve homodyne measurements of both observablesXL and PL by using, for example, a beam splitter. The non-commutativity of XL and PL leads to additional quantum noisebeing added during this procedure. The target atomic state has itsintrinsic quantum noise (coming from the Heisenberg uncertaintyrelations). All this extra noise leads to a limited fidelity for theclassical recording: for example, to a maximum fidelity of 50% forcoherent states5–7. Thus the challenge of implementing a quantummemory can be formulated as a faithful storing of the simul-taneously immeasurable values of XL and PL.

A number of quantum information protocols, such as eaves-dropping in quantum cryptography, quantum repeaters8, and linearoptics quantum computing9, would benefit from a memory meet-ing the following criteria: (1) the light pulse to be stored is sent by athird party in a state unknown to the memory party; (2) the state oflight is converted into a quantum state of the memory with a fidelityhigher than that of the classical recording. Several recent experi-ments10–13 have demonstrated entanglement of light and atoms.However, none of these experiments demonstrated memory obey-ing the two above criteria. In ref. 14, where squeezed light wasmapped onto atoms, the atomic state existed only while the lightwas on, so it was not a memory device. The electromagneticallyinduced transparency (EIT) approach has led to the demonstrationof a classical memory for light15,16. A theoretical proposal for EIT-based quantummemory for light has been published in ref. 3. Otherproposals for quantum memory for light with better-than-classicalquality of recording have also been published recently1–4.

Quantum state transfer from one species to another is mostsimply presented if both systems are described by canonicalquantum variables X,P. All canonical variables have the samecommutation relations and the same quantum noise for a givenstate, thus providing a common frame for the analysis of the statetransfer.

In the present work, the state of light is stored in the super-position of magnetic sublevels of the ground state of an atomicensemble. As in ref. 12, we introduce the operator J of the collectivemagnetic moment (orientation) of a ground state F. All atomicstates utilized here are not too far in phase space from the coherentspin state (CSS), for which only one projection has a non-zeromeanvalue, for example, kJxl ¼ J x, whereas the other two projections haveminimal quantum uncertainties, kdJ2yl¼ kdJ2z l¼

12 Jx: For all such

states, the commutator Jy; Jz

� �¼ iJx can be reduced to the canonical

commutator [XA,PA] ¼ i with XA ¼ Jy=ffiffiffiffiJx

p; PA ¼ Jz=

ffiffiffiffiJx

p: Hence

the y,z-components of the collective atomic angular momentumplay the role of canonical variables. Although the memory protocol,in principle, can work with a single atomic ensemble, experimentaltechnical noise is substantially reduced if two oppositely polarizedensembles placed in a bias magnetic field H are used (see Methods

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and Supplementary Methods for details). Combined canonicalvariables for two ensembles XA ¼ ðJy1 2 Jy2Þ=

ffiffiffiffiffiffiffi2Jx

p; PA ¼

ðJz1 þ Jz2Þ=ffiffiffiffiffiffiffi2Jx

pare then introduced, where Jx1 ¼2Jx2 ¼ Jx ¼

FNatoms: In the presence of H, the memory couples to the Q-sidebands of light: XL ¼

1ffiffiffiT

pÐ T

0 ðaþðtÞ þ aðtÞÞcosðQtÞdt; PL ¼

iffiffiffiT

pÐ T0 ða

þðtÞ2 aðtÞÞcosðQtÞdt; where Q is the Larmor frequency ofspin precession.

Quantum storage of light is achieved in three steps: (1) aninteraction of light with atoms; (2) a subsequent measurement ofthe transmitted light; and (3) feedback onto the atoms conditionedon the measurement result (Fig. 1). The off-resonant interaction oflight with spin polarized atomic ensembles has been describedelsewhere4,17–19, and is summarized in the Methods section. Theinteraction leads to the equations:

Xout

L ¼ Xin

L þ kPin

A ; Pout

L ¼ Pin

L

Xout

A ¼ Xin

A þ kPin

L ; Pout

A ¼ Pin

A

ð1Þ

These equations imply that light and atoms get entangled. Theremarkable simplicity of equations (1) provides a direct linkbetween an input light state, an atomic state, and an output light.Suppose the input light is in a vacuum (or in a coherent) state, andatoms are in a CSS with mean values kXLl¼ kXAl¼ kPLl¼ kPAl¼ 0and variances dX2

L ¼ dX2A ¼ dP2

L ¼ dP2A ¼ 1=2: The interaction par-

ameter k, whose value is crucial for the storage protocol, is thenreadily found as k2 ¼ 2 dXout

L

� �221:

For a perfect fidelity of mapping, the initial atomic state must bean entangled spin state such as in ref. 12, with dX2

A ! 0: The pulse tobe recorded, combined with the entangling pulse (see Methodssection), is sent through, and its variable Xout

L is measured.The measurement outcome, x ¼ Xin

L þ kPinA ; is fed back into the

atomic variable PA with a feedback gain g. The result is PmemA ¼

PinA 2 gx ¼ P

in

A ð12 kgÞ2 gXinL (see Supplementary Notes for a jus-

tification of this equation). With g ¼ k ¼ 1, the mapping of XinL

onto 2PmemA is perfect.

The second operator of light is already mapped onto atomsvia Xmem

A ¼ XinA þ Pin

L , see equation (1). For the entangled initialstate the mapping is perfect for this component too, Pin

L ! XmemA ;

leading to the fidelity of the light-to-atoms state transfer F ! 100%.If the initial atomic state is a CSS, the mapping is not perfect owingto the noisy operator Xin

A :However, fidelity F ¼ 82%, still markedlyhigher than the classical limit, can be achieved. Note that the abovediscussion holds for an arbitrary single mode input quantum stateof light.

In our experiment, the atomic storage unit consists of twosamples of caesium vapour placed in paraffin-coated glass cellsplaced inside magnetic shields (Fig. 1). H is applied along the x-direction with Q ¼ 322 kHz. Optical pumping along H initializesthe atoms in the first/second sample in the F ¼ 4, m F ¼ ^4 groundstate with the orientation above 99%. Hence Jx1 ¼2Jx2 ¼ Jx ¼4Natoms < 1:2£ 1012: We thoroughly check and regularly verifythat the initial spin state is close to CSS (Supplementary Methods).The coupling parameter k is varied by adjusting the density ofcaesium vapour.The input state a(t) is encoded in a 1-ms y-polarized pulse. The

state is chosen from the set {ainput} of coherent states with thephoton number in the range {knl ¼ 0, nmax} and an arbitrary phase.a(t) is generated as Q sidebands by an electro-optical modulator(EOM), and has the same spatial and temporal profile as the strongentangling field (more information can be found in the Methodssection). Thus the EOM plays the third party, providing the fieldto be stored. The pulses are detuned by 700MHz to the bluefrom the 6S1/2, F ¼ 4 ! 6P3/2, F ¼ 5 transition (l ¼ 852 nm).The polarization measurement of the light is followed by thefeedback onto atoms achieved by a 0.2ms radio-frequencymagneticpulse conditioned on the measurement result.The experimental verification of the quantum storage is then

carried out. A read-out x-polarized pulse is sent through thesamples with a delay of 0.7–10ms after the feedback is applied.Atomic memory generates a y-polarized pulse, which is analysed asfollows. As both Xmem

A and PmemA cannot be measured at the same

time, we carry out two series of measurements for each input state.Each series consists of 104 quantum storage sequences. To verify theXinL !2Pmem

A step of the storage, we measure the componentXread–outL ¼ Xread–in

L þ kPmemA of the read-out pulse (XL is a Stokes

parameter measured in units of shot noise, as discussed in theMethods section). An example of such a measurement carried outafter 0.7ms of storage is presented in Fig. 2a as a histogram of1k Xread–out

L (right histogram), with k measured as described in theMethods section and in Supplementary Methods. For this serieskPin

L l¼24 and kXinL l¼ 0; corresponding to knl ¼ 8 photons

in the pulse. From this measurement, we find the meankPmem

A l¼ 1k kX

read–outL l and the variance j2p ¼ k dPmem

A

� �2l¼

1k2 dXread–out

L

� �22 1

2

� �(see equations (1)) for the quantum state of

the memory. We note that only the knowledge of k and the shotnoise level of light is necessary for the determination of the meanvalues and variances of the atomic canonical variables from theexperimental data.

Figure 1 Experimental set-up. a, Atomic memory unit consisting of two caesium cells

inside magnetic shields 1 and 2. The path of the recorded and read-out light pulses is

shown with arrows. b, The simplified layout of the experiment. The input state of light with

the desired displacements X L, P L is generated with the electro-optic modulator (EOM).

The inset shows the pulse sequence for the quantum memory recording and read-out.

Pulse (1) is the optical pumping (4 ms), pulse (2) is the input light pulse a(t ) overlapped

with the strong entangling pulse in orthogonal polarization with amplitudeffiffiffiffiffiffiffiffinðt Þ

p: Pulse (3)

is the magnetic feedback pulse. Pulse (4) is the magnetic p/2 pulse used for the read out

of one of the atomic operators. Pulse (5) is the read-out optical pulse.

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Next, we run another series of storage with the same input statefor the verification of the step Pin

L ! XmemA : The Xmem

A operator doesnot couple to the read-out pulse in our geometry; therefore, wefirst apply a p/2-pulse (Fig. 1) to atoms converting Xmem

A ! PpA

and then measure PpA with the verifying pulse. We then find kXmem

A land j2x ¼ kðdXmem

A Þ2l of the memory state (left histogram).The above sequence is repeated for different input states. From

kPmemA l=kXin

L l and kXmemA l=kPin

L l; the mapping gains for the twoquadratures are determined. For the experimental data presentedin Figs 2 and 3a, these gains are 0.80 and 0.84 respectively, which isclose to the optimal gain for the chosen input set of states. This stepwould complete the proof of the classical memory performance,because we have shown that the y-polarized pulse recovered fromthe memory has the same mean amplitude and mean phase as theinput pulse (up to a chosen constant factor).To prove a quantum memory performance, we need in addition

to consider the quantum noise of the stored state. Towards this end,we plot the atomic variances j2p;j

2x for the storage time 0.7ms in

Fig. 3a. The experimentally obtained variances of the stored state areon average 33% below the best possible variance of the classicalrecording. Hence a density of stored states 33% higher than that forthe best classical recording can be obtained. Thus the goal of

quantum storage with less noise than for the classical recording isachieved.

Next, the overlap between the input state of light and the stateof the atomic memory is determined (Methods section). Anexample is shown in Fig. 2b. The fidelity F of the quantum recordingis then calculated for a given set {ainput}. For example, F ¼ð66:7^ 1:7Þ% for {ainput} ¼ {n ¼ 0 ! 8} and F ¼ (70.0 ^ 2.0)%for {ainput} ¼ {n ¼ 0 ! 4}, respectively, for the storage time of0.7ms. Note that the fidelity of the classical recording can exceed50% for a limited set {ainput}. The maximum classical fidelity for{ainput} ¼ {n ¼ 0 ! 8} is 55.4%, and for {ainput} ¼ {n ¼ 0 ! 4} itis 59.6%—still significantly lower than that for the quantumrecording.

The main sources of imperfection of our quantum memory aredecoherence of the atomic state and reflection off the cell walls. Wehave performed extensive studies of the atomic decoherence causedby the light-assisted collisional relaxation20 to optimize the fidelity.Figure 3b presents the fidelity of the stored state as a function of thestorage time. A simple model provides a good description for theobserved fidelity reduction.

The single observable read-out described above can be useful in,for example, quantum cryptography eavesdropping, where thememory is read after the basis has been publicly announced by

Figure 2 An example of the atomic memory performance. a, The input state of light in the

coherent state with kXLl¼ 0; kPLl¼24: The results of the read out of this state stored in

the atomic memory are shown as histograms of experimental realizations. The left/right

histogram shows the results for the X A/PA quadrature read out with/without the p/2-

pulse. Dotted gaussians represent the distributions for the best possible quantummemory

performance (fidelity 100%). b, The input coherent state of light (upper graph) and the

reconstructed state stored in the atomic memory (lower graph) for the input state as in a.

The reconstructed state is obtained from the results presented in a after subtracting the

noise of the read-out pulse.

Figure 3 Quantum noise of the stored state and the fidelity of quantum memory as a

function of time. a, Experimental and theoretical (quantum and classical) stored state

variances in atomic projection noise (PN) units. Triangles and filled circles are the

experimental variances for the atomic memory operators, denoted 2j2x and 2j2

p ;

respectively, in the text. Dash-dotted line, the fundamental boundary of three units of

noise between quantum and classical mapping for an arbitrary coherent input state5,6.

Dashed line, best classical variance for the experimental set of input states with photon

numbers between 0 and 8. Double-dot-dashed line, unity variance corresponding to

perfect mapping. b, Fidelity as a function of storage time for the set of states from 0 to 10

photons. Fidelity higher than the classical limit is maintained for up to 4 ms of storage.

Error bars are standard deviations.

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Alice and Bob. The present experiment also paves the way towardsthe proposed quantum cloning of light onto atomic memory21.However, other applications require complete state recovery viareverse mapping of the memory state onto light. Proposals forperforming this task within our approach have been published4,19,22.Probably the most intuitively clear protocol for the memory read-out is just to run the storage protocol presented here with reversedroles of light and atoms. Indeed, the equations of interaction (1) arecompletely symmetric. The read-out, as the storage, would involvethree steps: sending a read-out light pulse through atoms, measur-ing the spin projection X

out

A with an auxiliary light pulse, andapplying the feedback conditioned on this measurement to theread-out pulse.

In the present experiment, we have demonstrated thememory fora subset of linearly independent coherent states. Owing to thelinearity of quantum mechanics, this demonstration signifies thatour method provides faithful mapping for an arbitrary coherentstate. As any arbitrary quantum state can be written as a super-position of coherent states, our approach should in principle workfor an arbitrary quantum state, including entangled and singlephoton (qubit) states. A

MethodsQuantum coupling of light to two atomic ensembles in the presence of magneticfieldHere we discuss the physics behind the equations of interaction (1). The off-resonantatom/light interaction is described in terms of Stokes operators for the polarization stateof light and the collective spin of atoms4,17,18. The Stokes operators are defined as one halfof the photon number difference between orthogonal polarization modes: S1, betweenvertical x- and horizontal y-polarizations; S2, between the modes polarized at^458 to thevertical axis; and S3, between the left- and right-hand circular polarizations. In theexperiment, a strong entangling x-polarized pulse with photon flux n(t) is mixed on apolarizing beamsplitter with the y-polarized quantum field a(t) prior to interactionwith atoms. Hence the Stokes operators of the total optical field are S1ðtÞ ¼ S1ðtÞ ¼

12nðtÞ;

S2 ¼12

ffiffiffiffiffiffiffiffinðtÞ

pðaþðtÞþ aðtÞÞ; S3 ¼

i2

ffiffiffiffiffiffiffiffinðtÞ

pðaþðtÞ2 aðtÞÞ: Note that S2(t) and S3(t) are

proportional to the canonical variables for the quantum light mode, X ¼ 1ffiffi2

p ðaþðtÞþ aðtÞÞ;P ¼ iffiffi

2p ðaþðtÞ2 aðtÞÞ: Light is transmitted through the atomic samples placed in the

bias magnetic field oriented along the x-axis. The magnetic field allows for encoding of thememory at the Larmor frequency Q, thus dramatically reducing technical noise present atlow frequencies. However, in the presence of the Larmor precession, there is an undesiredcoupling of the single-cell variables Jy and Jz to each other. The introduction of the secondcell with the opposite Larmor precession allows us to introduce new two-cell variablesðJy1 2 Jy2Þ; ðJz1 þ Jz2Þ that do not couple to each other. As in ref. 12, where a similar trickwas used, the Stokes parameters of light transmitted through the two cells along the zdirection become:

Sout

2 ðtÞ ¼ Sin

2 ðtÞ þ aS1ðcosðQtÞ½Jz1 þ Jz2�þ sinðQtÞ½Jy1 þ Jy2�Þ; Sout

3 ðtÞ ¼ Sin

3 ðtÞ ð2Þ

where Jz,y are the projections in the frame rotating at Q, and a ¼ gl2

8pDA ; with g and l thenatural linewidth and the wavelength of the transition, respectively, D the detuning, and Athe beam cross-section. At the same time, the transverse spin components of the two cellsevolve as follows:

ddt ½Jz1 þ Jz2� ¼

ddt ½Jy1 þ Jy2� ¼ 0;

ddt ½Jy1 2 Jy2� ¼ 2aJxS

in

3 cosðQtÞ; ddt ½Jz1 2 Jz2� ¼ 2aJxS

in

3 sinðQtÞð3Þ

As evident from equation (3), in the process of propagation the operator Sin3 is recordedonto the operators Jy1 2 Jy2 and Jz1 2 Jz2 (the ‘back action’ of light on atoms via thedynamic Stark effect caused by light17,18), while the operators Jy1 þ Jy2 and Jz1 þ Jz2 are leftunchanged. The latter are read out onto Sout2 via the Faraday rotation, equation (2).

Canonical variables are defined for the quantum light mode as XL ¼1ffiffiffiT

pÐ T

0 ðaþðtÞ þ

aðtÞÞcosðQtÞdt; PL ¼iffiffiffiT

pÐ T

0 ðaþðtÞ2 aðtÞÞcosðQtÞdt; that is, the relevant lightmode involves

the Q-sidebands. T is the pulse duration, a(t) is normalized to the photon flux. XL and PL

(that is, S2 and S3) are detected by a polarization state analyser and by lock-in detection oftheQ component of the photocurrent. Note that the cos(Qt) component of light couples tothe ðJy1 2 Jy2Þ; ðJz1 þ Jz2Þ components of atomic storage variables (equations (2), (3)).The equivalent choice of a sin(Qt) modulation instead would mean the use of ðJy1 þ

Jy2Þ; ðJz1 2 Jz2Þ for the memory. The atomic canonical variables XA, PA are defined in themain section. With the above equations and definitions we straightforwardly deriveequations (1) under the assumption QT .. 1: Theoretically, the dimensionless couplingparameter in (1) is k2 ¼ 1

2a2Jx

ÐnðtÞdt:

Experimental calibration of the canonical variances for light and atomsCalculations of the fidelity, the gains, and the variances from the experimental data arebased on the experimental calibration of kdX2

Ll¼ kdP2Ll for the coherent (vacuum) state of

light and of kdX2Al¼ kdP2

Al for the CSS of atoms. The calibration for light is carried outalong the established procedure of determining the shot noise level for measurements of

S2, S3 with the quantum field in a vacuum state5,17. Variances and mean values for light arethenmeasured in units of this shot noise level. The calibration for the atomic CSS varianceis carried out with extreme care, and has shown excellent reproducibility (seeSupplementary Methods). As stated in the main text, as soon as the vacuum (shot) noiselevel for light is established and the atoms are in a CSS, the parameter k2 (equations (1)),important for calculations of atomic variances and fidelity, is easily determined ask2 ¼ 2 dXout

L

� �221: In the experiment, this is equivalent to k2 ¼

dSout2

� �22 dSin2� �2� �

= dSin2� �2

:

Fidelity and the state overlapTo calculate the fidelity of the transfer of an input coherent state into an output gaussianstate6, we first define an overlap function between an input state with mean values x1, p1

and the output state with the mean values and variances x2, p2, j2x ; j2p: Straightforward

integration yields:

O{x1;x2;p1;p2}¼ 2expð2ðx1 2 x2Þ2=ð1þ 2j2xÞ

2 ðp1 2 p2Þ2=ð1þ 2j2pÞÞ=

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 2j2x� �

1þ 2j2p

� �r

The fidelity of the transfer for a set of coherent states with mean amplitudes between a1

and a2 can then be found as an average overlap:

F ¼ p21 a22 2a2

1

� �21ð2p0

df

ða2

a1

O{a}ada

For classical recording from light onto atoms with gain g, the overlap between the inputcoherent state with the mean amplitude a¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 þ p2

pand the output state is given by

O{a}¼ ð1þ g2Þ21exp 2 12 ð12 gÞ2a2ð1þ g2Þ21

� �: The classical fidelity is then given by:

Fclass ¼ ðn2 2 n1Þ21ð12 gÞ22{expð2ð12 gÞ2n1ð1þ g2Þ21Þ

2 expð2ð12 gÞ2n2ð1þ g2Þ21Þ}

where we have introduced the mean photon number n ¼ 12a

2: F class ! 50% for arbitrarycoherent states when g ! 1. If a restricted class of coherent states is chosen as the input,F class . 50% can be obtained with a suitable choice of g. For a set of states analysed in themain text, {ainput} ¼ {n ¼ 0 ! 8}, the maximum classical fidelity of 55.4% is achievedwith a gain of 0.809.

Received 11 June; accepted 28 September 2004; doi:10.1038/nature03064.

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Supplementary Information accompanies the paper on www.nature.com/nature.

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NATURE |VOL 432 | 25 NOVEMBER 2004 | www.nature.com/nature 485© 2004 Nature Publishing Group

Acknowledgements We are grateful to N. Cerf and K. Hammerer for discussions. This research

was funded by the Danish National Research Foundation, by EU grants QUICOV, COVAQIAL

and CHIC, and by the project ‘Research Center for Optics’ of the Czech Ministry of Education.

I.C. and E.S.P. acknowledge the hospitality of the Institute for Photonic Sciences, Barcelona,

where part of this work was initiated.

Competing interests statement The authors declare that they have no competing financial

interests.

Correspondence and requests for materials should be addressed to E. P. (polzik@nbi.dk).

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Direct observation of the discretecharacter of intrinsic localizedmodes in an antiferromagnetM. Sato & A. J. Sievers

Laboratory of Atomic and Solid State Physics and Cornell Center for MaterialsResearch, Cornell University, Ithaca, New York 14850-2501, USA.............................................................................................................................................................................

In a strongly nonlinear discrete system, the spatial size of anexcitation can become comparable to, and influenced by, thelattice spacing. Such intrinsic localized modes (ILMs)—alsocalled ‘discrete breathers’ or ‘lattice solitons’—are responsiblefor energy localization in the dynamics of discrete nonlinearlattices1–5. Their energy profiles resemble those of localizedmodes of defects in a harmonic lattice but, like solitons, theycan move (although, unlike solitons, some energy is exchangedduring collisions between them). The manipulation of theselocalized energy ‘hotspots’ has been achieved in systems asdiverse as annular arrays of coupled Josephson junctions6,7,optical waveguide arrays8, two-dimensional nonlinear photoniccrystals9 and micromechanical cantilever arrays10. There is alsosome evidence for the existence of localized excitations in atomiclattices11–15, although individual ILMs have yet to be identified.Here we report the observation of countable localized excitationsin an antiferromagnetic spin lattice by means of a nonlinearspectroscopic technique. This detection capability permits theproperties of individual ILMs to be probed; the disappearance ofeach ILM registers as a step in the time-dependent signal, withthe surprising result that the energy staircase of ILM excitationsis uniquely defined.Rod-shaped samples of the quasi-one-dimensional antiferro-

magnet, (C2H5NH3)2CuCl4, are used in this study13,16–18 so thatthe antiferromagnetic resonance (AFMR) frequency occurs at thebottom of the spin-wave manifold. Mesoscale, high spin-precessionamplitude, magnetic ILMs can then be produced in the gap belowthis band because of the soft nonlinearity of the spin lattice. Asoutlined in Fig. 1a, there are four fundamental sequential time stepsrequired for these energy localization experiments: (1) a uniformspin wave mode of the biaxial antiferromagnet is driven to a largeprecession amplitude by an initial microwave pulse (frequency f1);(2) after an incubation period, the modulational instability of thelarge amplitude uniformmode takes hold, producingmany ILMs in abroad frequency band19; (3) a few of these ILMs are then locked20,21 tothe continuous wave (c.w.) middle power source f2; and, finally, (4)the experiment of interest, where a c.w. low power source f3 is used toproduce a mixing signal that depends on the number of ILMs in thelattice. After a spin-lattice relaxation time of T1 ¼ 1.5ms, thenumber of ILMs is determined by a quasi-steady state involving thef2 driver and the nonequilibrium AFMR13, and this number isexpected to decrease as this frequency difference increases.

Because the number of spins in an ILM is too small to be seen inabsorption, a nonlinear energy magnetometer has been developed,which automatically handles the four time steps for this lowtemperature ILM production and detection experiment. It relieson the third-order nonlinearity xð3Þ of the antiferromagnet22–24 tomake observable in nonlinear emission the small number of ILMsthat remain locked to the f2 driver, as illustrated in Fig. 1b. Non-linear and linear excitations in the same sample give very differentnonlinear responses: the signal produced by ILMs is enhanced,while that produced by the small precession amplitude extendedplane wave states, which are nearly harmonic, is suppressed. Thisnonlinear discrimination feature of the instrument brings the fewnanoscale ILMs out of the background of plane wave states forexperimental exploration.

Presented in Fig. 2a is the power spectrum produced at f3, where2f2 2 f3 ¼ fdet, and both f3 and fdet, the narrow band detectorfrequency, are scanned in tandem. Typical frequency positions forf1, f2 and the near-equilibrium AFMR are identified at the top ofFig. 2. The strong emission peak from the locked ILMs (see arrow) isobserved shifted to slightly lower frequencies from the f2 lockingoscillator (about 4MHz), while a somewhat weaker peak is shiftedup in frequency. (These sidebands represent a form of cross phasemodulation for the ILMs.) The logarithmic plot shown in Fig. 2bdisplays more clearly the other weaker spectral features, as well asthe excellent signal to noise level for these measurements. The weakpeak observed at 1.362GHz (see top right arrow) occurs when f3matches the AFMR frequency in a resonant four wave mixingprocess at T < 2ms. This weak feature at long times is independentof whether or not the f1 pulse is applied and, hence, is a property ofthe homogeneous solid. When ILMs are produced, they encompassstates originally associated with the spin wave band; however,because of the small number of ILMs studied (that is, only a fewILMs remain locked), 99.999% of the spins are in the low amplitudehomogeneous state. Figure 2b demonstrates that although the AFMRinvolves many spins, off resonance it is a very weak nonlinear mixer.

Figure 1 Schematic diagrams of the experimental procedure and the nonlinear process.

a, The chronological timing in the nonlinear energy magnetometer is shown for the

production and measurement of intrinsic localized modes (ILMs). AFMR,

antiferromagnetic resonance. b, Nonlinear mixing output for uniform excitations and for

ILMs. Large filled circles represent ILMs. Although small in number because of locking,

these strong nanoscale third order mixing elements dominate the output signal. Two

inputs are from the locking driver field H(f 2) and the remainder is from the probe field

H(f 3), giving a maximum output at 2f 2 2 f 3. Each ILM generates a nonlinear field H (3).

As the mixed signal power depends on the square of the total field, the square root of the

power is proportional to the number of ILMs.

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